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nLab vacuum stability (changes)

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Context

Vacua

vacuum

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

In quantum field theory, a given vacuum state is called stable if in a suitable sense it does not evolve into or from any orthogonal state.

In perturbative quantum field theory with specified S-matrix 𝒮(gS int)\mathcal{S}(g S_{int}) it makes sense to say that a vacuum state is stable if there is vanishing quantum probability for the vacuum state to scatter into a non-vacuum state, or for a non-vacuum state to scatter into the vacuum state. (e.g. Nikolic 01, p. 6)

Definition

Definition

(vacuum stability)

Let (E BV-BRST,L′,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) be a relativistic free vacuum according to (this def.), let 𝒮\mathcal{S} be a corresponding S-matrix scheme according to (this. def.), and let gS int∈LocObs(E BV-BRST)[[ℏ,g]]⟨g⟩g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle be a local observable, regarded as an adiabatically switched interaction action functional.

We say that the given Hadamard vacuum state (this prop.)

⟨−⟩:PolyObs(E BV-BRST) mc[[ℏ,g,j]]⟶ℂ[[ℏ,g,j]] \langle - \rangle \;\colon\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g, j ] ] \longrightarrow \mathbb{C}[ [ \hbar, g, j ] ]

is stable with respect to the interaction S intS_{int}, if for all elements of the Wick algebra

A∈PolyObs(E BV-BRST) mc[[ℏ,g]] A \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g] ]

we have

⟨A𝒮(gS int)⟩=⟨𝒮(gS int)⟩⟨A⟩AAandAA⟨𝒮(gS int) −1A⟩=1⟨𝒮(gS int)⟩⟨A⟩ \left\langle A \mathcal{S}(g S_{int}) \right\rangle \;=\; \left\langle \mathcal{S}(g S_{int}) \right\rangle \, \left\langle A \right\rangle \phantom{AA} \text{and} \phantom{AA} \left\langle \mathcal{S}(g S_{int})^{-1} A \right\rangle \;=\; \frac{1} { \left\langle \mathcal{S}(g S_{int}) \right\rangle } \left\langle A \right\rangle

Properties

Example

(scattering amplitudes as vacuum expectation values of interacting field observables)

Let (E BV-BRST,L′,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) be a relativistic free vacuum according to this def., let 𝒮\mathcal{S} be a corresponding S-matrix scheme according to this def., and let gS int∈LocObs(E BV-BRST)[[ℏ,g]]⟨g⟩g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle be a local observable regarded as an adiabatically switched interaction-functional, such that the vacuum state is stable with respect to gS intg S_{int} (def. 1).

Consider local observables

A in,1,⋯,A in,n in, A out,1,⋯,A out,n out∈LocObs(E BV-BRST)[[ℏ,g]] \array{ A_{in,1}, \cdots, A_{in , n_{in}}, \\ A_{out,1}, \cdots, A_{out, n_{out}} } \;\;\in\;\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]

whose spacetime support satisfies the following causal ordering:

A out,i out><A out,j outAAAA out,i out∨∧S int∨∧A in,i inAAAA in,i in><A in,j in A_{out, i_{out} } {\gt\!\!\!\!\lt} A_{out, j_{out}} \phantom{AAA} A_{out, i_{out} } {\vee\!\!\!\wedge} S_{int} {\vee\!\!\!\wedge} A_{in, i_{in}} \phantom{AAA} A_{in, i_{in} } {\gt\!\!\!\!\lt} A_{in, j_{in}}

for all 1≤i out<j out≤n out1 \leq i_{out} \lt j_{out} \leq n_{out} and 1≤i in<j in≤n in1 \leq i_{in} \lt j_{in} \leq n_{in}.

Then the vacuum expectation value of the Wick algebra-product of the corresponding interacting field observables (this def.) is

⟨(A out,1) int⋯(A out,n out) int(A in,1) int⋯(A in,n in) int⟩ =⟨A out,1⋯A out,n out|𝒮(gS int)|A in,1⋯A in,n in⟩ ≔1⟨𝒮(gS int)⟩⟨A out,1⋯A out,n out𝒮(gS int)A in,1⋯A in,n in⟩. \begin{aligned} & \left\langle {\, \atop \,} (A_{out, 1})_{int} \cdots (A_{out,n_{out}})_{int} \, (A_{in, 1})_{int} \cdots (A_{in,n_{in}})_{int} {\, \atop \,} \right\rangle \\ & = \left\langle {\, \atop \,} A_{out,1} \cdots A_{out,n_{out}} \right| \; \mathcal{S}(g S_{int}) \; \left| A_{in,1} \cdots A_{in, n_{in}} {\, \atop \,} \right\rangle \\ & \coloneqq \frac{1}{ \left\langle \mathcal{S}(g S_{int}) \right\rangle } \left\langle {\, \atop \,} A_{out,1} \cdots A_{out,n_{out}} \; \mathcal{S}(g S_{int}) \; A_{in,1} \cdots A_{in, n_{in}} {\, \atop \,} \right\rangle \,. \end{aligned}

These vacuum expectation values are interpreted, in the adiabatic limit where g sw→1g_{sw} \to 1, as scattering amplitudes (see this remark).

For proof see at S-matrix, this prop..

Examples

quantum probability theory – observables and states

References

Discussion for QED:

Discussion for Higgs field:

  • J.R. Espinosa, G. Giudice, A. Riotto, Cosmological implications of the Higgs mass measurement, JCAP 0805:002, 2008 (arXiv:0710.2484)

  • John Ellis, J.R. Espinosa, G.F. Giudice, A. Hoecker, A. Riotto, The Probable Fate of the Standard Model, Phys. Lett. B679:369-375, 2009 (arXiv:0906.0954)

  • Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, Gian Giudice, Filippo Sala, Alberto Salvio, Alessandro Strumia, Investigating the near-criticality of the Higgs boson (arXiv:1307.3536)

  • Anson Hook, John Kearney, Bibhushan Shakya, Kathryn M. Zurek, Probable or Improbable Universe? Correlating Electroweak Vacuum Instability with the Scale of Inflation, J. High Energ. Phys. (2015) 2015: 61 (arXiv:1404.5953)

  • Jose R. Espinosa, Gian F. Giudice, Enrico Morgante, Antonio Riotto, Leonardo Senatore, Alessandro Strumia, Nikolaos Tetradis, The cosmological Higgstory of the vacuum instability (arXiv:1505.04825)

  • William E. East, John Kearney, Bibhushan Shakya, Hojin Yoo, Kathryn M. Zurek, Spacetime Dynamics of a Higgs Vacuum Instability During Inflation, Phys. Rev. D 95, 023526 (2017) (arXiv:1607.00381)

  • Gordon Kane, Exciting Implications of LHC Higgs Boson Data (arXiv:1802.05199)

Last revised on February 8, 2020 at 10:50:02. See the history of this page for a list of all contributions to it.

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